Chapter 15

Verify that \(f\) gives a joint probability density function. Then find the expected values \(\mu_{X}\) and \(\mu_{Y}\) . $$ f(x, y)=\left\\{\begin{array}{ll}{6 x^{2} y,} & {\text { if } 0 \leq x \leq 1 \text { and } 0 \leq y \leq 1} \\ {0,} & {\text { otherwise. }}\end{array}\right. $$

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. \begin{equation} \int_{0}^{1} \int_{-\sqrt{1-y^{2}}}^{\sqrt{1-y^{2}}} 3 y d x d y \end{equation}

Converting to polar integrals a. The usual way to evaluate the improper integral \(I=\int_{0}^{\infty} e^{-x^{2}} d x\) is first to calculate its square: \(I^{2}=\left(\int_{0}^{\infty} e^{-x^{2}} d x\right)\left(\int_{0}^{\infty} e^{-y^{2}} d y\right)=\int_{0}^{\infty} \int_{0}^{\infty} e^{-\left(x^{2}+y^{2}\right)} d x d y\) Evaluate the last integral using polar coordinates and solve the resulting equation for \(I\). b. Evaluate \(\lim _{x \rightarrow \infty} \operatorname{erf}(x)=\lim _{x \rightarrow \infty} \int_{0}^{x} \frac{2 e^{-t^{2}}}{\sqrt{\pi}} d t\)

\(D\) is the prism whose base is the triangle in the \(x y\) -plane bounded by the \(x\) -axis and the lines \(y=x\) and \(x=1\) and whose top lies in the plane \(z=2-y .\)

Verify that \(f\) gives a joint probability density function. Then find the expected values \(\mu_{X}\) and \(\mu_{Y}\) . $$ f(x, y)=\left\\{\begin{array}{ll}{\frac{3}{2}\left(x^{2}+y^{2}\right),} & {\text { if } 0 \leq x \leq 1 \text { and } 0 \leq y \leq 1} \\ {0,} & {\text { otherwise. }}\end{array}\right. $$

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. \begin{equation} \int_{0}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} 6 x d y d x \end{equation}

\(\begin{aligned} \text { Converting to } & \text { a polar integral Evaluate the integral } \\ & \int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{\left(1+x^{2}+y^{2}\right)^{2}} d x d y \end{aligned}\)

Evaluate the integrals in Exercises \(41-44\) by changing the order of integration in an appropriate way. $$ \int_{0}^{1} \int_{0}^{1} \int_{x^{2}}^{1} 12 x z e^{z y^{2}} d y d x d z $$

Suppose that \(f\) is a uniform joint probability density function on \(0 \leq x < 2,0 \leq y < 3 .\) What is the formula for \(f ?\) What is the probability that \(X < Y ?\)

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. \begin{equation} \int_{1}^{e} \int_{0}^{\ln x} x y d y d x \end{equation}

Existence Integrate the function \(f(x, y)=1 /\left(1-x^{2}-y^{2}\right)\) over the disk \(x^{2}+y^{2} \leq 3 / 4 .\) Does the integral of \(f(x, y)\) over the disk \(x^{2}+y^{2} \leq 1\) exist? Give reasons for your answer.

Evaluate the spherical coordinate integrals in Exercises \(43-48\) $$ \int_{0}^{\pi} \int_{0}^{\pi} \int_{0}^{2 \sin \phi} \rho^{2} \sin \phi d \rho d \phi d \theta $$

Evaluate the integrals in Exercises \(41-44\) by changing the order of integration in an appropriate way. $$ \int_{0}^{1} \int_{\sqrt[3]{z}}^{1} \int_{0}^{\ln 3} \frac{\pi e^{2 x} \sin \pi y^{2}}{y^{2}} d x d y d z $$

The following formula defines a joint probability density function. What is the value of \(C ?\) What are the expected values \(\mu_{X}\) and \(\mu_{Y} ?\) $$ f(x, y)=\left\\{\begin{array}{ll}{C x y,} & {\text { if } 0 \leq x \leq 2 \text { and } 0 \leq y \leq 3} \\ {0,} & {\text { otherwise. }}\end{array}\right. $$

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. \begin{equation} \int_{0}^{\pi / 6} \int_{\sin x}^{1 / 2} x y^{2} d y d x \end{equation}

Area formula in polar coordinates Use the double integral in polar coordinates to derive the formula $$A=\int_{\alpha}^{\beta} \frac{1}{2} r^{2} d \theta$$ for the area of the fan-shaped region between the origin and polar curve \(r=f(\theta), \alpha \leq \theta \leq \beta\)

Evaluate the spherical coordinate integrals in Exercises \(43-48\) $$ \int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{0}^{2}(\rho \cos \phi) \rho^{2} \sin \phi d \rho d \phi d \theta $$

Evaluate the integrals in Exercises \(41-44\) by changing the order of integration in an appropriate way. $$ \int_{0}^{2} \int_{0}^{4-x^{2}} \int_{0}^{x} \frac{\sin 2 z}{4-z} d y d z d x $$

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$ \int_{0}^{3} \int_{1}^{e^{y}}(x+y) d x d y $$

Average distance to a given point inside a disk Let \(P_{0}\) be a point inside a circle of radius \(a\) and let \(h\) denote the distance from \(P_{0}\) to the center of the circle. Let \(d\) denote the distance from an arbitrary point \(P\) to \(P_{0} .\) Find the average value of \(d^{2}\) over the region enclosed by the circle. (Hint: Simplify your work by placing the center of the circle at the origin and \(P_{0}\) on the \(x\) -axis.)

Evaluate the spherical coordinate integrals in Exercises \(43-48\) $$ \int_{0}^{2 \pi} \int_{0}^{\pi} \int_{0}^{(1-\cos \phi) / 2} \rho^{2} \sin \phi d \rho d \phi d \theta $$

Finding an upper limit of an iterated integral Solve for \(a :\) $$ \int_{0}^{1} \int_{0}^{4-a-x^{2}} \int_{a}^{4-x^{2}-y} d z d y d x=\frac{4}{15} $$

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. \begin{equation} \int_{0}^{\sqrt{3}} \int_{0}^{\tan ^{-1} y} \sqrt{x y} d x d y \end{equation}

Area Suppose that the area of a region in the polar coordinate plane is $$A=\int_{\pi / 4}^{3 \pi / 4} \int_{\csc \theta}^{2 \sin \theta} r d r d \theta$$ Sketch the region and find its area.

Evaluate the spherical coordinate integrals in Exercises \(43-48\) $$ \int_{0}^{3 \pi / 2} \int_{0}^{\pi} \int_{0}^{1} 5 \rho^{3} \sin ^{3} \phi d \rho d \phi d \theta $$

Ellipsoid For what value of \(c\) is the volume of the ellipsoid \(x^{2}+(y / 2)^{2}+(z / c)^{2}=1\) equal to 8\(\pi ?\)

In Exercises \(47-56,\) sketch the region of integration, reverse the order of integration, and evaluate the integral. \begin{equation} \int_{0}^{\pi} \int_{x}^{\pi} \frac{\sin y}{y} d y d x \end{equation}

Evaluate the integral \(\iint_{R} \sqrt{x^{2}+y^{2}} d A,\) where \(R\) is the region inside the upper semicircle of radius 2 centered at the origin, but outside the circle \(x^{2}+(y-1)^{2}=1\)

Evaluate the spherical coordinate integrals in Exercises \(43-48\) $$ \int_{0}^{2 \pi} \int_{0}^{\pi / 3} \int_{\sec \phi}^{2} 3 \rho^{2} \sin \phi d \rho d \phi d \theta $$

Sketch the region of integration, reverse the order of integration, and evaluate the integral. \begin{equation} \int_{0}^{2} \int_{x}^{2} 2 y^{2} \sin x y d y d x \end{equation}

Evaluate the integral \(\iint_{R}\left(x^{2}+y^{2}\right)^{-2} d A,\) where \(R\) is the region inside the circle \(x^{2}+y^{2}=2\) for \(x \leq-1 .\)

Maximizing a triple integral What domain \(D\) in space maximizes the value of the integral $$ \iiint_{D}\left(1-x^{2}-y^{2}-z^{2}\right) d V ? $$ Give reasons for your answer.

Evaluate the spherical coordinate integrals in Exercises \(43-48\) $$ \int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{0}^{\sec \phi}(\rho \cos \phi) \rho^{2} \sin \phi d \rho d \phi d \theta $$

Sketch the region of integration, reverse the order of integration, and evaluate the integral. \begin{equation} \int_{0}^{1} \int_{y}^{1} x^{2} e^{x y} d x d y \end{equation}

In Exercises \(49-52,\) use a CAS to change the Cartesian integrals into an equivalent polar integral and evaluate the polar integral. Perform the following steps in each exercise. $$ \begin{array}{l}{\text { a. Plot the Cartesian region of integration in the } x y \text { -plane. }} \\ {\text { b. Change each boundary curve of the Cartesian region in part }} \\ {\text { (a) to its polar representation by solving its Cartesian equation for } r \text { and } \theta .}\end{array} $$ $$ \begin{array}{l}{\text { c. Using the results in part (b), plot the polar region of integra- }} \\ {\text { tion in the } r \theta \text { -plane. }} \\\ {\text { d. Change the integrand from Cartesian to polar coordinates. De- }} \\ {\text { termine the limits of integration from your coordinates. De- }} \\ {\text { evaluate the polar integral using the CAS integration utility. }}\end{array} $$ $$ \int_{0}^{1} \int_{x}^{1} \frac{y}{x^{2}+y^{2}} d y d x $$

In Exercises \(49-52,\) use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. $$ \begin{array}{l}{F(x, y, z)=x^{2} y^{2} z \quad \text { over } \quad \text { the solid cylinder bounded by }} \\ {x^{2}+y^{2}=1 \text { and the planes } z=0 \text { and } z=1}\end{array} $$

The previous integrals suggest there are preferred orders of integration for spherical coordinates, but other orders give the same value and are occasionally easier to evaluate. Evaluate the integrals in Exercises \(49-52\) . $$ \int_{0}^{2} \int_{-\pi}^{0} \int_{\pi / 4}^{\pi / 2} \rho^{3} \sin 2 \phi d \phi d \theta d \rho $$

Sketch the region of integration, reverse the order of integration, and evaluate the integral. \begin{equation} \int_{0}^{2} \int_{0}^{4-x^{2}} \frac{x e^{2 y}}{4-y} d y d x \end{equation}

In Exercises \(49-52,\) use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. $$ \begin{array}{l}{F(x, y, z)=|x y z| \text { over the solid bounded below by the }} \\ {\text { paraboloid } z=x^{2}+y^{2} \text { and above by the plane } z=1}\end{array} $$

The previous integrals suggest there are preferred orders of integration for spherical coordinates, but other orders give the same value and are occasionally easier to evaluate. Evaluate the integrals in Exercises \(49-52\) . $$ \int_{\pi / 6}^{\pi / 3} \int_{\csc \phi}^{2 \csc \phi} \int_{0}^{2 \pi} \rho^{2} \sin \phi d \theta d \rho d \phi $$

Sketch the region of integration, reverse the order of integration, and evaluate the integral. \begin{equation} \int_{0}^{2 \sqrt{\ln 3}} \int_{y / 2}^{\sqrt{\ln 3}} e^{x^{2}} d x d y \end{equation}

In Exercises \(49-52,\) use a CAS to change the Cartesian integrals into an equivalent polar integral and evaluate the polar integral. Perform the following steps in each exercise. $$ \begin{array}{l}{\text { a. Plot the Cartesian region of integration in the } x y \text { -plane. }} \\ {\text { b. Change each boundary curve of the Cartesian region in part }} \\ {\text { (a) to its polar representation by solving its Cartesian equation for } r \text { and } \theta .}\end{array} $$ $$ \begin{array}{l}{\text { c. Using the results in part (b), plot the polar region of integra- }} \\ {\text { tion in the } r \theta \text { -plane. }} \\\ {\text { d. Change the integrand from Cartesian to polar coordinates. De- }} \\ {\text { termine the limits of integration from your coordinates. De- }} \\ {\text { evaluate the polar integral using the CAS integration utility. }}\end{array} $$ $$ \int_{0}^{1} \int_{-y / 3}^{y / 3} \frac{y}{\sqrt{x^{2}+y^{2}}} d x d y $$

In Exercises \(49-52,\) use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. $$ \begin{array}{l}{F(x, y, z)=\frac{z}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}} \text { over the solid bounded below by }} \\ {\text { the cone } z=\sqrt{x^{2}+y^{2}} \text { and above by the plane } z=1}\end{array} $$

The previous integrals suggest there are preferred orders of integration for spherical coordinates, but other orders give the same value and are occasionally easier to evaluate. Evaluate the integrals in Exercises \(49-52\) . $$ \int_{0}^{1} \int_{0}^{\pi} \int_{0}^{\pi / 4} 12 \rho \sin ^{3} \phi d \phi d \theta d \rho $$

Sketch the region of integration, reverse the order of integration, and evaluate the integral. \begin{equation} \int_{0}^{3} \int_{\sqrt{x / 3}}^{1} e^{y^{3}} d y d x \end{equation}

In Exercises \(49-52,\) use a CAS to change the Cartesian integrals into an equivalent polar integral and evaluate the polar integral. Perform the following steps in each exercise. $$ \begin{array}{l}{\text { a. Plot the Cartesian region of integration in the } x y \text { -plane. }} \\ {\text { b. Change each boundary curve of the Cartesian region in part }} \\ {\text { (a) to its polar representation by solving its Cartesian equation for } r \text { and } \theta .}\end{array} $$ $$ \begin{array}{l}{\text { c. Using the results in part (b), plot the polar region of integra- }} \\ {\text { tion in the } r \theta \text { -plane. }} \\\ {\text { d. Change the integrand from Cartesian to polar coordinates. De- }} \\ {\text { termine the limits of integration from your coordinates. De- }} \\ {\text { evaluate the polar integral using the CAS integration utility. }}\end{array} $$ $$ \int_{0}^{1} \int_{y}^{2-y} \sqrt{x+y} d x d y $$

In Exercises \(49-52,\) use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. $$ \begin{array}{l}{F(x, y, z)=x^{4}+y^{2}+z^{2} \text { over the solid sphere } x^{2}+y^{2}+} \\ {z^{2} \leq 1}\end{array} $$

The previous integrals suggest there are preferred orders of integration for spherical coordinates, but other orders give the same value and are occasionally easier to evaluate. Evaluate the integrals in Exercises \(49-52\) . $$ \int_{\pi / 6}^{\pi / 2} \int_{-\pi / 2}^{\pi / 2} \int_{\csc \phi}^{2} 5 \rho^{4} \sin ^{3} \phi d \rho d \theta d \phi $$

Sketch the region of integration, reverse the order of integration, and evaluate the integral. \begin{equation} \int_{0}^{1 / 16} \int_{y^{1 / 4}}^{1 / 2} \cos \left(16 \pi x^{5}\right) d x d y \end{equation}

Sketch the region of integration, reverse the order of integration, and evaluate the integral. \begin{equation} \int_{0}^{8} \int_{\sqrt[3]{x}}^{2} \frac{d y d x}{y^{4}+1} \end{equation}